Afterthoughts on Modeling
The goal of modeling is to gain insight into some problem that occurs in the natural world of real events. In a few cases there is sufficient data and enough understanding of the actual processes that it makes sense to test the model against numbers obtained from the field and to make predictions that can be verified in a relatively unambiguous manner, as in the crab model of Chapter 1. In other cases this is not possible, because a trustworthy database either is too meager or doesn’t exist or, as is the case in the restricted-access fishery model of Chapter 8, because an understanding of the underlying dynamics is inadequate. In the latter case, a model can still be useful, however, in providing a plausible metaphor for the observable behavior and even to suggest what the actual poorly known dynamics might be. The model can then be tested in some gross manner and provide direction for future investigations. Both kinds of models appear in this book, as you have seen. Although most problems discussed here are temporal in nature, spatial considerations sometimes cannot be avoided, as in Chapters 10 and 11.
Certain themes recur in different settings in various chapters, showing that a common problem can be viewed from varying angles, depending on the questions asked. Even if they are handled by different methods there is at least an undercurrent of commonality. An example is the behavior of a blood-clotting cascade in Chapter 9 and the algae bloom problem of Chapter 10. Each describes an excitable system that is activated when certain parameters exceed a certain threshold. In one case, the equations describe reactions among cellular organisms in a spatial setting; in the other, they describe biochemical reactions taking place over time. A common thread links the two phenomena. Strengthening this bond are the measles-epidemic and viral-contamination models of Chapter 9 as well as the model for the onset of a brown-tide outbreak, all of which also display threshold behavior. And so the fact that in each case a similar differential equation model was put to work should not be too much of a surprise.
We also saw this commonality in Chapter 10, in terms of traveling-wave solutions to three seemingly dissimilar problems. Moreover, the idea that diffusive instability can induce spatial patterning, the core topic of Chapter 11, is not unlike the notion that temporal instability can induce cyclic patterns, as we discovered in Chapters 8 and 9. In each instance an interplay of activating and inhibiting reactions together with diffusion sufficed to provide the required stimulus.
Some of these models entailed fairly sophisticated arguments for their formulation, and the surprising thing is that relatively straightforward mathematics was employed to elucidate their complexities. In truth, not much about differential equations was needed, and even less was required about partial differential equations. Elaborate solution techniques were not called for, so a basic background in these topics sufficed. The same applies to the probabilistic and statistical methods that we employed, and that is part of the good news regarding the models that I chose to include.
Multiple and conflicting objectives in societal problems provide another example of commonality. Issues of this type came up in Chapters 2, 3, 4, and 8. In each instance, fairly simple optimization schemes were employed to penetrate the problems. In retrospect, this is again not unexpected.
Though most of the models are deterministic in nature, stochastic tools occasionally come into play, as in Chapters 3, 5, 6, and 7. In Chapter 5 we encountered Bayesian thinking that, outside of the community of professional statisticians, is not as well known among workers in the social and biological sciences as it might be. The same is true about the lack of familiarity with power laws because of the longstanding role of the normal distribution in these fields. That is why three additional chapters on probabilistic models were included in this current edition of the book.
Depending on the questions you ask, one formulation of a problem may be more appropriate than another; if one is to make any headway at all it is wise to ask modest questions at first to avoid being overwhelmed. In this book, simplifying assumptions are made in each chapter in the hope that the phenomenon we are looking at is robust enough to exhibit behavior that is qualitatively similar to the real thing, even though it has been stripped of much of its complexity. We need to be aware of our assumptions and take them into account when we attempt to interpret what the models tell us. I try to do that in this book, but inevitably there are hidden factors that one is only dimly conscious of, and this can lead to misleading conclusions. That, unfortunately, is one of the pitfalls of modeling.
We strived to let the mathematics be driven by the problems, not the other way around. That is why each chapter begins with a description of the problem setting and then follows this with a suggestion for a likely approach. Good models begin this way, and to do otherwise is to engage in an exercise of trying to tailor a problem to fit the mathematics. This is another modeling pitfall that should be avoided.
Finally, the reader may have been struck by the large number of references to articles that have appeared in the New York Times, The New Yorker, Scientific American, and the like, supporting or explaining in layperson’s terms the results of social and biological investigations that were at least partially based on the kind of modeling discussed in this book. It is a vindication of the relevance of these models to a wide assortment of issues of genuine concern.